Theme Essential Question: How are functions used to describe relationships that happen in our lives?
Essential Questions: How do you write an equation for a function given a table, graph, or description?
Standards
8. F.3.Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
8. F.4.Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
Objectives
The student will write linear equations from a table.
The student will write linear equations from a graph.
The student will write linear equations from a word problem.
Reflection and/or Comments from your PCSSD 8th Grade Curriculum Team In this lesson students should be able to work from all representations (verbal, table, graph, or equation). They will require ample opportunities to move from one representation to another. It is important that they communicate how each representation is interconnected and how to use each representation.
Background Information Recommended: For a quick overview of the standard(s) to be addressed in this lesson, see Arizona’s Content Standards Reference Materials. **http://www.azed.gov/wp-content/uploads/PDF/MathGr8.pdf**
(Taken from **Ohio Dept of Education Mathematics Model Curriculum 6-28-2011** )
Distance time problems are notorious in mathematics. In this cluster, they serve the purpose of illustrating how the rates of two objects can be represented, analyzed and described in different ways: graphically and algebraically. Emphasize the creation of representative graphs and the meaning of various points. Then compare the same information when represented in an equation.
Assessment Product
Students will continue creating a portfolio detailing their growth pertaining to the topics addressed throughout this unit on functions. The Entry provided follows the theme of construction repair and remodel
Entry #4
A plumber charges $25 fee for coming to fix a leaky faucet and $10 for each hour he works.
Create a table that shows how much they pay for each of the first 10 hours.
Create a graph with the information from the table.
Write the equation representing the plumber charge.
Explain how this equation is obtained from the table.
Explain how this equation is obtained from the graph.
Complete your explanation, by showing how the two ways of finding the equation (table and graph) are related.
What is the meaning of slope in this problem?
What is the meaning of the y-intercept in this problem?
Answer the following questions:
Use the table, graph, and equation to determine the cost of the plumber working 15 hours.
If the plumber works for 25 hours, use the table, graph, and equation to find how much he will charge.
Key Questions
What is the process to write a linear equation from a table?
What is the process to write a linear equation from a graph?
What is the process to write a linear equation from a word problem?
How do you interpret the rate of change (slope) and initial value (y-intercept) in terms of the situation it models?
Observable Student Behaviors Given a table, graph, or a word problem that produces a constant rate of change, the student can write a linear equation.
Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.
Vocabulary Input, output, domain, range, independent and dependent variable, linear equation, linear function, rate of change, unit rate, constant rate of change or constant, slope, slope-intercept form, derive
Suggested Activities [see Legend to highlight MCO and HYS]
Houghton Mifflin OnCore Mathematics Middle School Grade 8 Unit 2-5 p.49-52
ABC Mastering the Common Core in Mathematics Chapter 5-9 and 5-14, p. 57-65 (with extensions)
Gizmo Lessons
Linear Functions
Determine if a relation is a function from the mapping diagram, ordered pairs, or graph. Use the graph to determine if it is linear. (Student Exploration Sheet & Teacher Guide Available)
Slope-Intercept Form of a line-Activity A
Compare the slope‑intercept form of a linear equation to its graph. Vary the coefficients and explore how the graph changes in response.
Points, Lines, and Equations
Compare the graph of a linear function to its rule and to a table of its values. Change the function by dragging two points on the line. Examine how the rule and table change. (Student Exploration Sheet & Teacher Guide Available)
Distance-Time Graph
Create a graph of a runner's position versus time and watch the runner complete a 40-yard dash based on the graph you made. Notice the connection between the slope of the line and the speed of the runner. What will the runner do if the slope of the line is zero? What if the slope is negative? Add a second runner (a second graph) and connect real-world meaning to the intersection of two graphs. (Student Exploration Sheet & Teacher Guide Available)
Modeling Linear Systems-Activity A
Experiment with a system of two lines representing a cat‑and‑mouse chase. Adjust the speeds of the cat and mouse and the head start of the mouse, and immediately see the effects on the graph and on the chase. Connect real‑world meaning to slope, y‑intercept, and the intersection of lines.
Simple and Compound Interest
Find the current balance and the interest charged on an investment using the graph of the interest function and by directly calculating.
Teaching the Common Core Math Standards with Hands-On Activities by Judith Muschla
8.F.3 p. 189
8.F.4 Activity 1p. 191, Activity 2 p. 192
JBHM Grade 8 GP2, Unit 4, p. 297-328
Glencoe Pre Algebra Chapter 8, Lesson 7, p. 404-408
Grade: 8 Unit: 3 Week: 5 Dates: 1/7-1/11
Content: Writing Equations to Describe Functions
Theme Essential Question:
How are functions used to describe relationships that happen in our lives?
Essential Questions:
How do you write an equation for a function given a table, graph, or description?
Standards
Objectives
Reflection and/or Comments from your PCSSD 8th Grade Curriculum Team
In this lesson students should be able to work from all representations (verbal, table, graph, or equation). They will require ample opportunities to move from one representation to another. It is important that they communicate how each representation is interconnected and how to use each representation.
Background Information
Recommended: For a quick overview of the standard(s) to be addressed in this lesson, see Arizona’s Content Standards Reference Materials.
**http://www.azed.gov/wp-content/uploads/PDF/MathGr8.pdf**
(Taken from **Ohio Dept of Education Mathematics Model Curriculum 6-28-2011** )
Distance time problems are notorious in mathematics. In this cluster, they serve the purpose of illustrating how the rates of two objects can be represented, analyzed and described in different ways: graphically and algebraically. Emphasize the creation of representative graphs and the meaning of various points. Then compare the same information when represented in an equation.
Assessment
Product
- A plumber charges $25 fee for coming to fix a leaky faucet and $10 for each hour he works.
- Create a table that shows how much they pay for each of the first 10 hours.
- Create a graph with the information from the table.
- Write the equation representing the plumber charge.
- Explain how this equation is obtained from the table.
- Explain how this equation is obtained from the graph.
- Complete your explanation, by showing how the two ways of finding the equation (table and graph) are related.
- What is the meaning of slope in this problem?
- What is the meaning of the y-intercept in this problem?
Answer the following questions:Key Questions
Observable Student Behaviors
Given a table, graph, or a word problem that produces a constant rate of change, the student can write a linear equation.
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Vocabulary
Input, output, domain, range, independent and dependent variable, linear equation, linear function, rate of change, unit rate, constant rate of change or constant, slope, slope-intercept form, derive
Suggested Activities [see Legend to highlight MCO and HYS]
- Houghton Mifflin OnCore Mathematics Middle School Grade 8 Unit 2-5 p.49-52
- ABC Mastering the Common Core in Mathematics Chapter 5-9 and 5-14, p. 57-65 (with extensions)
- Gizmo Lessons
- Linear Functions
Determine if a relation is a function from the mapping diagram, ordered pairs, or graph. Use the graph to determine if it is linear. (Student Exploration Sheet & Teacher Guide Available)- Slope-Intercept Form of a line-Activity A
Compare the slope‑intercept form of a linear equation to its graph. Vary the coefficients and explore how the graph changes in response.- Points, Lines, and Equations
Compare the graph of a linear function to its rule and to a table of its values. Change the function by dragging two points on the line. Examine how the rule and table change. (Student Exploration Sheet & Teacher Guide Available)- Distance-Time Graph
Create a graph of a runner's position versus time and watch the runner complete a 40-yard dash based on the graph you made. Notice the connection between the slope of the line and the speed of the runner. What will the runner do if the slope of the line is zero? What if the slope is negative? Add a second runner (a second graph) and connect real-world meaning to the intersection of two graphs. (Student Exploration Sheet & Teacher Guide Available)- Modeling Linear Systems-Activity A
Experiment with a system of two lines representing a cat‑and‑mouse chase. Adjust the speeds of the cat and mouse and the head start of the mouse, and immediately see the effects on the graph and on the chase. Connect real‑world meaning to slope, y‑intercept, and the intersection of lines.- Simple and Compound Interest
Find the current balance and the interest charged on an investment using the graph of the interest function and by directly calculating.- Teaching the Common Core Math Standards with Hands-On Activities by Judith Muschla
- 8.F.3 p. 189
- 8.F.4 Activity 1p. 191, Activity 2 p. 192
- JBHM Grade 8 GP2, Unit 4, p. 297-328
- Glencoe Pre Algebra Chapter 8, Lesson 7, p. 404-408
- Glencoe Algebra I Chaper 5, Lesson 4 , p.280-285
- Extension, Point-Slope form, p.286-291
- Extension, Parallel & Perpendicular, p. 292-297
Highly Recommended:http://illustrativemathematics.org/illustrations/552 (F.4)
http://illustrativemathematics.org/illustrations/477 (F.4)
http://illustrativemathematics.org/illustrations/584 (F.4)
http://illustrativemathematics.org/illustrations/383 (F.4)
http://illustrativemathematics.org/illustrations/247 (F.4)
The Illustrative Mathematics Project offers guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students will experience in a faithful implementation of the Common Core State Standards. The website features a clickable version of the Common Core in mathematics and the first round of "illustrations" of specific standards with associated classroom tasks and solutions.
Diverse Learners
Odyssey (teacher discretion)
Skills Tutor (teacher discretion)
Algebra’scool: Unit C Module 9
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